Critical speed or whirling speed of a drive shaft is such a speed at which the deflection of the driving shaft increases drastically due to the effect of the cetripetal force of the rotating cardon shaft and the shaft starts whirling.
In this part of the automobile propeller shaft design calculation tutorial we will see the critical speed calculation procedure for the example problem.
The calculation goes as below:

Calculate maximum static deflection of the driveshaft (δ) : The max. Static deflection is calculated from the equation
δ=(5*m*g*cosθ*L^3)/(384*E*J) = 0.036579 mm
Where,
m – mass of the propeller shaft = 18.65 kg (refer part1)
g – acceleration due to gravity = 9.8 m/s2
θ – Inclinition of the cardon shaft = 2 degree (for our example)
L – length of the shaft = 1250 mm (for our example)
E – Youngs modulus = 207000 Mpa (for our example)
J – Polar moment of inertia of the shaft = 613592.3 mm4 (refer part1)

Calculate the critical speed of the shaft (Nc): The whirling speed of the shaft can be calculated from the equation
Nc=[30*sqrt(g)]/[pi()*sqrt(δ)] = 4942 rpm
Where,
g – acceleration due to gravity = 9.8 m/s2
pi() =3.14
δ – maximum static deflection of the driveshaft = 0.036579 mm (for our example)
From the example problem statement it is seen that the speed of the drive shaft is 6500 rpm, which is much higher than the calculated critical speed of shaft (4942 rpm), hence the drive shaft design is not safe. To increase the critical speed of the cardon shaft we have to reduce the max. Static deflection and that can be achieved either by using a same diameter hollow (lighter) shaft of changing the material properties (i.e., changing the young’s modulus).